The session was chaired by UN envoy Ismail Ould Cheikh Ahmed and attended by four members of each side, one source close to the talks, who spoke to Anadolu Agency on condition of anonymity due to restrictions on speaking to media, said.
"The government delegation insisted that Yemeni state institutions must be surrendered [from Houthi control] and Houthi militiamen must withdraw from cities occupied earlier," the source said.
"The Houthi delegation, for its part, insisted on the establishment of a unity government in which they would play a role," the source added.
Immediately after Saturday's session, Abdul-Malik al-Mikhlafi, head of the government delegation, described the ongoing gap between the two sides' positions as "wide".
In a series of tweets, he declared that the Houthis "only want power; they are demanding [a role in] the government and power-sharing".
He added: "I'm surprised how this militia – which turned against the state and destroyed its institutions, laws and army – can now demand a role in power-sharing".
The government delegation insists that it represents Yemen's sole legitimate governing authority, citing last year's UN Security Council Resolution 2216.
The Houthis, for their part, say they represent the country's de facto rulers, having run sovereign state institutions – including its central bank – for more than one year.
Sources close to the talks told Anadolu Agency that UN envoy Ould Ahmed had called for the formation of "national security groups" at the provincial level, to be followed by the signing of a comprehensive peace agreement aimed at ending the fighting, which has continued in Yemen despite a ceasefire.
The same sources had told Anadolu Agency earlier that the main sticking point in the talks hinged on "the legitimacy of President Abd Rabbuh Mansour Hadi and his government".
The Houthi delegation, sources say, want to see Hadi replaced with a "presidential transitional council" to temporarily run the country's affairs.
This would be followed by the replacement of the current government – led by Prime Minister Ahmed Obaid bin Daghr – with a "unity government" to include Houthi representatives.
The government, meanwhile, the same sources said, insist that the Houthis first surrender their weapons and withdraw from cities occupied earlier.
Only then would the current government be expanded to include representatives of the Houthis and their allies.
Hadi would then preside over a "transitional phase" that would end with the amendment of Yemen's constitution and new presidential elections.
Yemen has been racked by chaos since 2014, when the Houthis and their allies overran capital Sanaa and several other parts of the country, forcing Hadi and his Saudi-backed government to temporarily flee to Riyadh.
In March of last year, Saudi Arabia and its Arab allies launched a massive military campaign in Yemen aimed at reversing Houthi gains and restoring Hadi's embattled government.
According to UN figures, the ongoing conflict has led to the death of some 6,400 Yemenis to date and forced some 2.5 million to flee their homes.
On April 11, UN-brokered peace talks kicked off in Kuwait following the announcement of a ceasefire.
The Yemeni government delegation at the talks demands the immediate implementation of UNSC Resolution 2216, which calls on the Houthis to withdraw from cities occupied earlier and to lay down their arms.
Ali Abo-Rezeg contributed to this report from Ankara.